The Hopkins Symptoms Checklist

Items

Questionnaire items

Descriptive statistics

Full sample

	C14yr (N=20095)
SCL01
N	20015
1	15634 (78.1%)
2	2699 (13.5%)
3	1105 (5.5%)
4	577 (2.9%)
SCL02
N	20047
1	14189 (70.8%)
2	4174 (20.8%)
3	1168 (5.8%)
4	516 (2.6%)
SCL04
N	20032
1	8918 (44.5%)
2	7524 (37.6%)
3	2633 (13.1%)
4	957 (4.8%)
SCL07
N	19961
1	12605 (63.1%)
2	5067 (25.4%)
3	1672 (8.4%)
4	617 (3.1%)
SCL17
N	20000
1	10707 (53.5%)
2	5703 (28.5%)
3	2359 (11.8%)
4	1231 (6.2%)
SCL18
N	19941
1	14609 (73.3%)
2	3623 (18.2%)
3	1244 (6.2%)
4	465 (2.3%)
SCL22
N	20002
1	10694 (53.5%)
2	6027 (30.1%)
3	2306 (11.5%)
4	975 (4.9%)
SCL24
N	20010
1	8848 (44.2%)
2	6596 (33.0%)
3	3013 (15.1%)
4	1553 (7.8%)
SCL79
N	19986
1	15067 (75.4%)
2	3003 (15.0%)
3	1157 (5.8%)
4	759 (3.8%)

Genomic sample

	C14yr (N=14554)
SCL01
N	14495
1	11365 (78.4%)
2	1937 (13.4%)
3	800 (5.5%)
4	393 (2.7%)
SCL02
N	14517
1	10311 (71.0%)
2	3055 (21.0%)
3	793 (5.5%)
4	358 (2.5%)
SCL04
N	14507
1	6453 (44.5%)
2	5522 (38.1%)
3	1869 (12.9%)
4	663 (4.6%)
SCL07
N	14462
1	9145 (63.2%)
2	3709 (25.6%)
3	1196 (8.3%)
4	412 (2.8%)
SCL17
N	14484
1	7845 (54.2%)
2	4085 (28.2%)
3	1713 (11.8%)
4	841 (5.8%)
SCL18
N	14433
1	10643 (73.7%)
2	2599 (18.0%)
3	870 (6.0%)
4	321 (2.2%)
SCL22
N	14484
1	7787 (53.8%)
2	4385 (30.3%)
3	1635 (11.3%)
4	677 (4.7%)
SCL24
N	14484
1	6372 (44.0%)
2	4821 (33.3%)
3	2210 (15.3%)
4	1081 (7.5%)
SCL79
N	14470
1	10949 (75.7%)
2	2171 (15.0%)
3	820 (5.7%)
4	530 (3.7%)

Distributions

Correlations

Table

Plots

Full

C14yr

IRT models

Model comparisons

Model fit (best)

Model fit assessed using M2 index (C2 variant), which is specifically designed to assess the fit of item response models for ordinal data. The C2 variant is mainly useful when polytomous response models do not have sufficient degrees of freedom to compute M2. This function also computes associated fit indices that are based on fitting the null model. The M2-based root mean square error of approximation is the primary fit index. The standardized root mean square residual (SRMSR) and comparative fit index (CFI) is used to assess adequacy of model fit.The RMSEA and SRMSR can be used to suggest that the data fit the model, with suggested cutoff values of RMSEA <= .06 and SRMSR <= .08

Item fit

To assess how well each item fits the model the recommended index is S-X2 (and it’s RMSEA value), which can be used to assess degree of item fit. Values less than .06 are considered evidence of adequate fit.

Item fit 2

Proponents of the “Rasch Model” (see further below) often rather report infit and outfit statistics. We can get those by adding the argument fit_stats = “infit”). We get both mean-squared and standardized versions of these measures (Linacre provides some guidelines to interpret these: https://www.rasch.org/rmt/rmt162f.htm). Roughly speaking the non-standardized values should be between .5 and 1.5 to not be degrading.

Item fit 2 plots

SclAnxie | Q11C

Note: Items with values within 0.5 and 1.5 are considered to be productive for measurement.

SclDepre | Q11C

Note: Items with values within 0.5 and 1.5 are considered to be productive for measurement.

SclFull | Q11C

Note: Items with values within 0.5 and 1.5 are considered to be productive for measurement.

Person fit

SclAnxie | Q11C

If a person with a high theta (that is high latent ability) answers does not answer an easy item correctly, this person does not fit the model. Conversely, if a person with a low ability answer a very difficult question correctly, it likewise doesn’t fit the model. In practice, there will most likely be a few people who do not fit the model well. But as long as the number of non-fitting respondents is low, we are good. We mostly look again at infit and outfit statistics. If less than 5% of the respondents have higher or lower infit and outfit values than 1.96 and -1.96, we are good.

SclDepre | Q11C

If a person with a high theta (that is high latent ability) answers does not answer an easy item correctly, this person does not fit the model. Conversely, if a person with a low ability answer a very difficult question correctly, it likewise doesn’t fit the model. In practice, there will most likely be a few people who do not fit the model well. But as long as the number of non-fitting respondents is low, we are good. We mostly look again at infit and outfit statistics. If less than 5% of the respondents have higher or lower infit and outfit values than 1.96 and -1.96, we are good.

SclFull | Q11C

If a person with a high theta (that is high latent ability) answers does not answer an easy item correctly, this person does not fit the model. Conversely, if a person with a low ability answer a very difficult question correctly, it likewise doesn’t fit the model. In practice, there will most likely be a few people who do not fit the model well. But as long as the number of non-fitting respondents is low, we are good. We mostly look again at infit and outfit statistics. If less than 5% of the respondents have higher or lower infit and outfit values than 1.96 and -1.96, we are good.

IRT parameters

There are two assessments of item-latent trait relationships. The graded response model parameterization generates discrimination and difficulty (location) parameters. The slope parameter (a-parameter) is a measure of how well a [item] category discriminates respondents with different levels of the latent trait. Larger values, or steeper slopes, are better at differentiating theta. A slope also can be interpreted as an indicator of the strength of a relationship between a [item] category and latent trait, with higher slope values corresponding to stronger relationships. Two location parameters (b-parameters) are listed for each item. Location parameters are interpreted as the value of theta that corresponds to a .5 probability of responding at or above that location on an item. There are m-1 location parameters where m refers to the number of response categories on the response scale.

For the nominal response model, the a-parameters are not the same as the graded response model discrimination parameters, but instead represent the parameter ordering (with one discrimination parameter for each category). Under Bock’s (1972) original formulation, both sets of parameters were constrained to sum to 1 for proper identification and interpretation, so some of the a-parameters will be negative. Lower values of a-parameters represent lower categories (where the rank of the values indicate the empirical ordering), while lower values of the c-parameter (the intercept) represent the relative ‘easiness’ of the category to select. A positive a parameter for a category indicates that, in general, the probability of responding in that category increases as the trait level increases, though there may be some trait ranges where the probability decreases (e.g., the probability of responding in the second highest category, with an a parameter of 0.825, increases throughout much of the theta range, but eventually decreases at the highest thetas). The a parameters are not necessarily in ascending order in the nominal model, nor are they necessarily at equal intervals. Here the c-parameter (i.e. b-parameter) represents 50% of this vs reference category

Factor loadings

Factor loadings can be interpreted as a strength of the relationship between an item and the latent variable (F1). The Communalities (h2) are squared factor loadings and are interpreted as the variance accounted for in an item by the latent trait. A substantive relationship between item and with the latent trait is regarded as a loading > .50

Reliability

Cat. characteristic curves

SclAnxie | Q11C

[[1]]

The probabilities of responding to specific categories in an item’s response scale are graphically displayed in the category characteristic curves. Symmetrical curves represent the probability of endorsing a response category (P1-P3). These curves have a functional relationship with theta. As theta increases, the probability of endorsing a category increases and then decreases as responses transition to the next higher category. Each curve represents the probability of selecting a particular response option as a function of the latent trait. In the figure, each item has three curves representing the three response options. The response categories are labeled as “P1” to “P3”. For all items, the curves follow the same order as the response categories.

SclDepre | Q11C

[[1]]

The probabilities of responding to specific categories in an item’s response scale are graphically displayed in the category characteristic curves. Symmetrical curves represent the probability of endorsing a response category (P1-P3). These curves have a functional relationship with theta. As theta increases, the probability of endorsing a category increases and then decreases as responses transition to the next higher category. Each curve represents the probability of selecting a particular response option as a function of the latent trait. In the figure, each item has three curves representing the three response options. The response categories are labeled as “P1” to “P3”. For all items, the curves follow the same order as the response categories.

SclFull | Q11C

[[1]]

The probabilities of responding to specific categories in an item’s response scale are graphically displayed in the category characteristic curves. Symmetrical curves represent the probability of endorsing a response category (P1-P3). These curves have a functional relationship with theta. As theta increases, the probability of endorsing a category increases and then decreases as responses transition to the next higher category. Each curve represents the probability of selecting a particular response option as a function of the latent trait. In the figure, each item has three curves representing the three response options. The response categories are labeled as “P1” to “P3”. For all items, the curves follow the same order as the response categories.

Item info curves

SclAnxie | Q11C

[[1]]

Information is a statistical concept that refers to the ability of an item to accurately estimate scores on theta. Item level information clarifies how well each item contributes to score estimation precision with higher levels of information leading to more accurate score estimates. In polytomous models, the amount of information an item contributes depends on its slope parameter—–the larger the parameter, the more information the item provides. The further apart the location parameters (b-parameters), the more information the item provides. Typically, an optimally informative polytomous item will have a large location and broad category coverage (as indicated by location parameters) over theta. Information functions are best illustrated by the item information curves for each item as displayed above. These curves show that item information is not a static quantity, rather, it is conditional on levels of theta. The relationship between slopes and information is illustrated here. The “wavy” form of the curves reflects the fact that item information is a composite of category information, that is, each category has an information function which is then combined to form the item information function.

SclDepre | Q11C

[[1]]

Information is a statistical concept that refers to the ability of an item to accurately estimate scores on theta. Item level information clarifies how well each item contributes to score estimation precision with higher levels of information leading to more accurate score estimates. In polytomous models, the amount of information an item contributes depends on its slope parameter—–the larger the parameter, the more information the item provides. The further apart the location parameters (b-parameters), the more information the item provides. Typically, an optimally informative polytomous item will have a large location and broad category coverage (as indicated by location parameters) over theta. Information functions are best illustrated by the item information curves for each item as displayed above. These curves show that item information is not a static quantity, rather, it is conditional on levels of theta. The relationship between slopes and information is illustrated here. The “wavy” form of the curves reflects the fact that item information is a composite of category information, that is, each category has an information function which is then combined to form the item information function.

SclFull | Q11C

[[1]]

Information is a statistical concept that refers to the ability of an item to accurately estimate scores on theta. Item level information clarifies how well each item contributes to score estimation precision with higher levels of information leading to more accurate score estimates. In polytomous models, the amount of information an item contributes depends on its slope parameter—–the larger the parameter, the more information the item provides. The further apart the location parameters (b-parameters), the more information the item provides. Typically, an optimally informative polytomous item will have a large location and broad category coverage (as indicated by location parameters) over theta. Information functions are best illustrated by the item information curves for each item as displayed above. These curves show that item information is not a static quantity, rather, it is conditional on levels of theta. The relationship between slopes and information is illustrated here. The “wavy” form of the curves reflects the fact that item information is a composite of category information, that is, each category has an information function which is then combined to form the item information function.

Scale info and cond. SE

SclAnxie | Q11C

[[1]]

Information for individual items can be summed to form a scale information function. A scale information function is a summary of how well items, overall, provide statistical information about the latent trait. Further, scale information values can be used to compute conditional standard errors which indicate how precisely scores can be estimated across different values of theta. The relationship between scale information and conditional standard errors is illustrated above. The solid blue line represents the scale information function. The overall scale provides the most information at the peak of the blue line. The red line provides a visual reference about how estimate precision varies across theta with smaller values corresponding to better estimate precision. Conditional standard errors mathematically mirror the scale information curve.

SclDepre | Q11C

[[1]]

Information for individual items can be summed to form a scale information function. A scale information function is a summary of how well items, overall, provide statistical information about the latent trait. Further, scale information values can be used to compute conditional standard errors which indicate how precisely scores can be estimated across different values of theta. The relationship between scale information and conditional standard errors is illustrated above. The solid blue line represents the scale information function. The overall scale provides the most information at the peak of the blue line. The red line provides a visual reference about how estimate precision varies across theta with smaller values corresponding to better estimate precision. Conditional standard errors mathematically mirror the scale information curve.

SclFull | Q11C

[[1]]

Information for individual items can be summed to form a scale information function. A scale information function is a summary of how well items, overall, provide statistical information about the latent trait. Further, scale information values can be used to compute conditional standard errors which indicate how precisely scores can be estimated across different values of theta. The relationship between scale information and conditional standard errors is illustrated above. The solid blue line represents the scale information function. The overall scale provides the most information at the peak of the blue line. The red line provides a visual reference about how estimate precision varies across theta with smaller values corresponding to better estimate precision. Conditional standard errors mathematically mirror the scale information curve.

Cond. reliability

SclAnxie | Q11C

[[1]]

The concept of conditional reliability is illustrated in the above. This curve is mathematically related to both scale information and conditional standard errors through transformations. It also is possible to compute a single reliability estimate for the model (displayed in a table above).

SclDepre | Q11C

[[1]]

The concept of conditional reliability is illustrated in the above. This curve is mathematically related to both scale information and conditional standard errors through transformations. It also is possible to compute a single reliability estimate for the model (displayed in a table above).

SclFull | Q11C

[[1]]

The concept of conditional reliability is illustrated in the above. This curve is mathematically related to both scale information and conditional standard errors through transformations. It also is possible to compute a single reliability estimate for the model (displayed in a table above).

Scale characteristic curve

SclAnxie | Q11C

[[1]]

A latent trait scoring procedure called expected a posteriori (EAP) estimation was used to generate factor scores. IRT model-based scores reflect the impacts of parameter estimates obtained from the IRT model used. As a result, because they are weighted by item parameters, theta score estimates often show more variability than summed scores. They also can be interpreted in the standard normal framework; because they are given in a standard normal metric, we can use our knowledge of the standard normal distribution to make score comparisons across individuals. The scale characteristic function can be graphically displayed as shown above. It has a straightforward use; for any given estimated theta score we can easily find a corresponding expected true score in the summed scale score metric. These true score transformations often are of interest in practical situations where scale users are not familiar with theta scores. Also, true score estimates can be used in other important statistical analyses and are often improvements over traditional summed scores.

SclDepre | Q11C

[[1]]

A latent trait scoring procedure called expected a posteriori (EAP) estimation was used to generate factor scores. IRT model-based scores reflect the impacts of parameter estimates obtained from the IRT model used. As a result, because they are weighted by item parameters, theta score estimates often show more variability than summed scores. They also can be interpreted in the standard normal framework; because they are given in a standard normal metric, we can use our knowledge of the standard normal distribution to make score comparisons across individuals. The scale characteristic function can be graphically displayed as shown above. It has a straightforward use; for any given estimated theta score we can easily find a corresponding expected true score in the summed scale score metric. These true score transformations often are of interest in practical situations where scale users are not familiar with theta scores. Also, true score estimates can be used in other important statistical analyses and are often improvements over traditional summed scores.

SclFull | Q11C

[[1]]

A latent trait scoring procedure called expected a posteriori (EAP) estimation was used to generate factor scores. IRT model-based scores reflect the impacts of parameter estimates obtained from the IRT model used. As a result, because they are weighted by item parameters, theta score estimates often show more variability than summed scores. They also can be interpreted in the standard normal framework; because they are given in a standard normal metric, we can use our knowledge of the standard normal distribution to make score comparisons across individuals. The scale characteristic function can be graphically displayed as shown above. It has a straightforward use; for any given estimated theta score we can easily find a corresponding expected true score in the summed scale score metric. These true score transformations often are of interest in practical situations where scale users are not familiar with theta scores. Also, true score estimates can be used in other important statistical analyses and are often improvements over traditional summed scores.

Factor scores

Descriptive statistics

Full sample

	C14yr (N=20095)
Anxie
N	20093
Mean (SD)	0.000 (0.868)
Median	-0.111
Range	-0.922 - 2.752
Depre
N	20087
Mean (SD)	-0.002 (0.880)
Median	-0.051
Range	-1.048 - 2.624
Full
N	20095
Mean (SD)	-0.003 (0.915)
Median	-0.005
Range	-1.227 - 2.952

Genomic sample

	C14yr (N=14554)
Anxie
N	14553
Mean (SD)	-0.006 (0.858)
Median	-0.111
Range	-0.922 - 2.752
Depre
N	14546
Mean (SD)	-0.010 (0.871)
Median	-0.051
Range	-1.048 - 2.624
Full
N	14554
Mean (SD)	-0.010 (0.905)
Median	-0.005
Range	-1.227 - 2.952

Distributions

Correlations

Table

Plot

Cross-sectional models

Descriptives

	x
Males	0.46
ChAge_mean	14.11
ChAge_sd	0.13
MoAge_mean	45.34
MoAge_sd	4.31
FaAge_mean	47.67
FaAge_sd	5.06

Mixed models

C14yr

	Anxie_C14yr			Depre_C14yr			Full_C14yr
Predictors	Estimates	CI	p	Estimates	CI	p	Estimates	CI	p
(Intercept)	-0.89	-0.93 – -0.85	<0.001	-0.86	-0.90 – -0.82	<0.001	-0.98	-1.03 – -0.94	<0.001
ChSex	0.58	0.55 – 0.60	<0.001	0.55	0.53 – 0.58	<0.001	0.63	0.61 – 0.66	<0.001
ChAge	-0.01	-0.03 – 0.00	0.052	-0.01	-0.02 – 0.01	0.224	-0.01	-0.03 – 0.00	0.112
MoAge	-0.02	-0.04 – -0.00	0.036	-0.01	-0.02 – 0.01	0.595	-0.01	-0.03 – 0.01	0.235
FaAge	0.01	-0.01 – 0.03	0.176	0.01	-0.01 – 0.03	0.474	0.01	-0.01 – 0.03	0.238
Random Effects
σ2	0.58			0.54			0.59
τ00	0.07 FamID			0.14 FamID			0.12 FamID
ICC	0.11			0.20			0.17
N	13792 FamID			13792 FamID			13792 FamID
Observations	14483			14476			14484
Marginal R2 / Conditional R2	0.112 / 0.212			0.101 / 0.282			0.122 / 0.272

Quantile rank scores

Distributions

Correlations

Table

Plot

Qrs x Zsc

Anxie_C14yr

Depre_C14yr

Full_C14yr

Mixed models

C14yr

	Anxie_C14yr			Depre_C14yr			Full_C14yr
Predictors	Estimates	CI	p	Estimates	CI	p	Estimates	CI	p
(Intercept)	-0.68	-0.73 – -0.63	<0.001	-0.49	-0.54 – -0.44	<0.001	-0.27	-0.32 – -0.21	<0.001
ChSex	0.44	0.41 – 0.47	<0.001	0.31	0.28 – 0.34	<0.001	0.16	0.13 – 0.20	<0.001
ChAge	-0.01	-0.02 – 0.01	0.446	-0.01	-0.02 – 0.01	0.399	0.00	-0.02 – 0.02	0.983
MoAge	-0.02	-0.05 – -0.00	0.026	-0.02	-0.04 – 0.01	0.132	-0.02	-0.04 – 0.00	0.077
FaAge	0.01	-0.01 – 0.03	0.356	0.01	-0.01 – 0.04	0.205	0.01	-0.02 – 0.03	0.646
Random Effects
σ2	0.87			0.87			0.93
τ00	0.05 FamID			0.07 FamID			0.03 FamID
ICC	0.05			0.07			0.03
N	13792 FamID			13792 FamID			13792 FamID
Observations	14483			14476			14484
Marginal R2 / Conditional R2	0.051 / 0.100			0.025 / 0.097			0.007 / 0.041